The converse is likewise demonstrated in this figure. Let the Rectangle of ae, and ay, be equall to the quadrate of ai.
I say, that ai doth touch the circle. For let, by the [26 e], ao the tangent be drawne: Item let au, ui, and uo bee drawne. Here the oblong of ea, and ay, is equall to the quadrate of ao, by the [27 e]: And to the quadrate of ai, by the grant. Therefore ai, and ao, are equall. Then is uo, by the [20 e], perpendicular to the tangent. Here the triangles auo, and aui, are equilaters: And by the [1 e vij], equiangles. But the angle at o is a right angle: Therefore also a right angle and equall to it is that at i, by the [13 e iij], wherefore ai is perpendicular to the end of the diameter: And, by the [19 e], it toucheth the periphery.
Therefore
28. All tangents falling from the same point are equall.
Or, Touch lines drawne from one and the same point are equall: H.
Because their quadrates are equall to the same oblong.
And
29. The oblongs made of any secant from the same point, and of the outter segment of the secant are equall betweene themselves. Camp. 36 p iij.
The reason is because to the same thing.
And